Difficulty: Introductory to Intermediate | Prerequisites: Budget line basics (how to plot a budget line from income and prices, what the slope means). Familiarity with indifference curves is helpful for the rational choice section.
This set of notes covers three extensions of the basic budget-line model. First, how the rational choice framework explains shifts in preferences (not just income or prices) using indifference curves. Second, how real-world pricing schemes create non-linear, kinked budget constraints rather than the clean straight lines of the textbook baseline. Third, how goods consumed in fixed proportions (perfect complements) produce L-shaped indifference curves and a distinctive solution method. Together these move you from drawing simple budget lines to handling the kinds of problems that actually show up on exams.
Preference shifts (not just price or income changes) can move the optimal bundle by reshaping indifference curves. Real-world pricing plans often produce kinked budget lines with different slopes in different segments. When goods must be consumed in a fixed ratio (perfect complements), the consumer's optimum sits at the corner of L-shaped indifference curves, and you solve by plugging the fixed ratio into the budget constraint.
Rational choice model
The framework in which a consumer maximises utility subject to a budget constraint. The optimal bundle is where the highest indifference curve is tangent to the budget line.
In simple terms, people pick the best they can afford.
Preference shift
A change in a consumer's tastes that reshapes their indifference map (the family of indifference curves). This is distinct from a change in income or prices, which moves the budget line but leaves preferences unchanged.
Think of it as the map of "what I like" being redrawn, while the budget line stays put.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to trade one good for another while remaining equally satisfied. Graphically, it is the absolute value of the slope of the indifference curve at a given point.
Kinked budget constraint
A budget line with one or more kinks (slope changes) caused by pricing structures that charge different rates at different quantities. The line is piecewise linear rather than a single straight line.
In simple terms, the "exchange rate" between goods changes once you pass a threshold.
Composite good
A theoretical good representing "everything else" the consumer might buy, typically priced at $1 per unit so that spending on it equals the number of units. It lets you reduce a many-goods world to a two-good graph.
Perfect complements
Goods that a consumer always wants in a fixed proportion (e.g. 4 parts coffee to 1 part milk). The indifference curves are L-shaped, with the corner lying along the line that represents the fixed ratio.
Think of left shoes and right shoes: an extra left shoe with no matching right shoe adds nothing.
Fixed-proportion consumption
The pattern where additional units of one good provide no extra satisfaction unless matched by units of the other good in the required ratio. The optimal bundle always sits at the corner of the L-shaped indifference curve.
The assignment asks: if the political process reflects people's wishes, what in the rational choice model explains increased demand for security after 9/11?
The budget line does not move. National income did not suddenly increase. The budget constraint between "security goods" (horizontal axis) and "all other goods" (vertical axis) stays in the same position.
Preferences changed. The 9/11 attacks made people value security more relative to other goods. In the model, this means the indifference curves reshaped: they became steeper, reflecting a higher marginal rate of substitution of other goods for security. People became willing to give up more of other goods to get an additional unit of security.
The new tangency point shifts rightward. With steeper indifference curves meeting the same budget line, the optimal bundle moves to the right along the budget line, toward more security and less of other goods.
Graphically:
Draw the budget line from the vertical intercept (all income on other goods) to the horizontal intercept (all income on security).
Draw the original indifference curve tangent to the line at a point weighted toward other goods.
Draw a new, steeper indifference curve tangent to the same line at a point further right, showing more security consumption.
The key insight is that the rational choice model can explain changes in consumption even when neither income nor prices change, through a shift in the shape of the indifference map.
The problem: income is $12/wk, composite good costs $1/unit (so spending on the composite good equals units of it). Two telephone plans:
Plan A: $0.05 per call, no fixed fee
Budget equation: Composite good + 0.05 × Calls = 12
If zero calls: 12 units of composite good. If zero composite good: 12 / 0.05 = 240 calls.
This is a straight line from (0, 12) to (240, 0) with slope –0.05.
Plan B: $2/wk flat fee, first 30 calls free, then $0.05 per call beyond 30
The $2 flat fee is paid regardless, leaving $10 for the composite good and calls beyond 30.
Segment 1 (0 to 30 calls): Each call costs nothing. The consumer has $10 for composite good (income minus the $2 fee). This segment is a horizontal line at Composite good = 10, running from 0 to 30 calls.
Segment 2 (beyond 30 calls): Each additional call costs $0.05. Budget equation for this segment: Composite good + 0.05 × (Calls – 30) = 10, which simplifies to Composite good = 10 – 0.05 × (Calls – 30). At 30 calls, composite good = 10. At maximum calls (0 composite good): Calls = 30 + (10 / 0.05) = 230.
The budget line under Plan B has a kink at 30 calls: flat from 0 to 30, then sloping downward at –0.05 from 30 onward.
Comparing the two plans graphically:
For 0 to 40 calls, Plan A gives more composite good (no flat fee). At 0 calls under Plan A you have 12 units; under Plan B you have only 10.
For a high volume of calls, Plan B catches up because the first 30 are free. Plan B's sloped segment starts at (30, 10), while Plan A at 30 calls gives 12 – 0.05 × 30 = 10.5. They cross at 40 calls, where both yield 10 units of composite good.
Beyond 40 calls Plan B dominates: both segments slope at –0.05, but Plan B's segment sits above Plan A's.
The consumer's effective budget constraint is the outer envelope of the two plans.
Boris wants coffee and milk in a fixed 4:1 ratio. His budget is $9/wk. Coffee costs $1/oz, milk costs $0.50/oz.
Setting up the fixed ratio: Let M = ounces of milk. Then coffee = 4M (four times as much coffee as milk).
Budget constraint: 1 × (4M) + 0.50 × M = 9, which gives 4M + 0.5M = 9, so 4.5M = 9, therefore M = 2 oz of milk and coffee = 4 × 2 = 8 oz.
Weekly purchase at original prices: 8 oz coffee, 2 oz milk. Total cost: 8 × $1 + 2 × $0.50 = $9. The entire budget is spent.
When the price of coffee rises to $3.25/oz:
New budget constraint: 3.25 × (4M) + 0.50 × M = 9, which gives 13M + 0.5M = 9, so 13.5M = 9, therefore M = 9/13.5 = 2/3 oz of milk and coffee = 4 × (2/3) = 8/3 ≈ 2.67 oz.
Boris buys roughly 2.67 oz of coffee and 0.67 oz of milk. Both quantities fall because the goods are consumed in a locked ratio: when one becomes more expensive, the consumer must cut both.
Graphical interpretation:
The indifference curves are L-shaped, with corners along the ray Coffee = 4 × Milk (a line from the origin with slope 4 on the coffee axis, or equivalently slope 1/4 if milk is on the horizontal axis).
The original budget line crosses this ray at (2, 8) when milk is on the horizontal axis.
When coffee's price rises, the budget line pivots inward (steeper), and the new intersection with the ray moves closer to the origin at approximately (0.67, 2.67).
The optimal bundle is always at the corner of the L, on the ray. There is no substitution between the goods, only an income effect.
Perfect complements utility function:
U(X, Y) = \min\left(\frac{X}{a},\; \frac{Y}{b}\right)where the consumer wants goods in the ratio a:b. For Boris, X = coffee, Y = milk, a = 4, b = 1.
Solving for the optimal bundle with perfect complements:
Set the ratio: X = (a/b) × Y. For Boris: Coffee = 4 × Milk.
Substitute into the budget constraint: Px × (a/b) × Y + Py × Y = I.
Solve for Y, then find X.
Kinked budget constraint (Plan B telephone example):
Segment 1 (0 to 30 calls):
Y = I - F = 12 - 2 = 10 \quad \text{(composite good, calls free)}Segment 2 (beyond 30 calls):
Y = (I - F) - P_c \times (\text{Calls} - 30)where F = flat fee, Pc = per-call price.
Kinked budget constraints appear everywhere: mobile phone plans with included minutes, gym memberships with pay-per-class add-ons, electricity tariffs with block pricing, and streaming services with ad-free upgrades. Any pricing structure that charges differently above or below a threshold produces a kink.
Perfect complements describe goods that are useless without their pair: printer ink and printers, nuts and bolts in manufacturing, or specific input ratios in chemical processes. The 4:1 coffee-to-milk ratio is a simplified version of how a barista might think about a latte.
Students often assume that increased demand for security after 9/11 must mean the budget line shifted. It did not. The explanation is a preference shift (indifference curves changed shape), not a budget change.
With kinked budget constraints, students sometimes draw a single straight line and ignore the kink. The whole point of the problem is that the slope changes at a threshold. Draw each segment separately.
For perfect complements, students often try to set MRS equal to the price ratio (the tangency condition). This does not work because L-shaped indifference curves have no smooth tangent at the corner. Instead, use the fixed-ratio condition directly.
Students sometimes forget that with perfect complements, a price increase in one good reduces consumption of both goods, not just the one that got more expensive. There is no substitution effect, only an income effect.
⚠️ The preference-shift question (9/11 security) is a common essay or short-answer prompt. The examinable skill is explaining a consumption change that comes from preferences, not from a budget shift.
⚠️ Kinked budget constraint problems test whether you can draw piecewise-linear budget lines and identify the kink point correctly. Get the coordinates of the kink right.
⚠️ Perfect complements problems test a specific solution technique (substitute the ratio, not the MRS). Know the method cold.
⚠️ Expect a question asking you to compare two pricing plans graphically and identify the crossover point where one plan becomes cheaper than the other.
True or False: After 9/11, the rational choice model explains increased security spending through a shift in the budget line.
True or False: Under Plan B in the telephone problem, the first 30 calls cost $0.05 each.
Fill in the blank: With perfect complements, the indifference curves are _____ shaped.
True or False: When the price of coffee rises and coffee and milk are perfect complements, Boris reduces his consumption of coffee but keeps milk the same.
Fill in the blank: The kink in Plan B's budget constraint occurs at _____ calls.
Q: Using the rational choice model, explain why consumers might demand more security spending after a terrorist attack, even if national income and prices have not changed.
A: The attack shifts preferences. Indifference curves become steeper (higher MRS of other goods for security), meaning people are willing to sacrifice more of other goods for each additional unit of security. With the budget line unchanged, the new tangency point lies further to the right on the security axis. Consumption shifts toward security and away from other goods.
Q: Under Plan A ($0.05/call, no fee) with $12 income, what are the intercepts of the budget line?
A: Vertical intercept (composite good) = 12. Horizontal intercept (calls) = 12 / 0.05 = 240.
Q: Under Plan B ($2 fee, 30 free calls, then $0.05/call), what are the coordinates of the kink point?
A: The kink is at 30 calls and 10 units of composite good (income $12 minus the $2 fee, with no per-call charges yet).
Q: Boris consumes coffee and milk in a 4:1 ratio. Coffee is $1/oz, milk is $0.50/oz, and his budget is $9/wk. How much of each does he buy?
A: Let M = milk. Coffee = 4M. Budget: 1(4M) + 0.50(M) = 9 → 4.5M = 9 → M = 2 oz milk, Coffee = 8 oz.
Q: If coffee rises to $3.25/oz, how does Boris's consumption change?
A: New budget: 3.25(4M) + 0.50(M) = 9 → 13.5M = 9 → M = 2/3 oz milk, Coffee = 8/3 ≈ 2.67 oz. Both fall because perfect complements allow no substitution; the entire adjustment is an income effect.
Q: Why can you not use the MRS = price ratio condition to solve a perfect complements problem?
A: L-shaped indifference curves have no smooth tangent at the corner. The MRS is undefined there (zero on one arm, infinite on the other). Instead, use the fixed-ratio condition directly and substitute into the budget constraint.
Preference shifts connect to welfare economics and public choice theory: how society decides to allocate resources when collective preferences change. Kinked budget constraints lead into the study of non-linear pricing and quantity discounts, which appear again in industrial organisation. Perfect complements are one endpoint of the substitution spectrum; the other is perfect substitutes (linear indifference curves), and most exam questions will test both extremes alongside the standard Cobb-Douglas case.
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